Double and multiple sampling plans were invented to give a questionable lot another chance. For example, if  in double sampling the results of the first sample are not conclusive with regard to accepting or rejecting, a second sample is taken. Application of double sampling requires that a first sample of size n₁ is taken at random from the (large) lot. The number of defectives is then counted and compared to the first sample's acceptance number a₁ and rejection number r₁  Denote the number of defectives in sample 1 by d₁ and in sample 2 by d₂ , then: 
 

The required parameters to  construct the OC curve are similar to the single sample case.  The two points of interest are (p₁, 1-a) and (p₂,b), where p₁ is the lot fraction defective for plan 1 and p₂ is the lot fraction defective for plan 2. As far as the respective sample sizes is concerned, the second sample size must be equal to or an even multiple of the first sample size. 

There exist a variety of tables that assist the user in constructing double and multiple sampling plans. The index to these tables is the p₂/p₁ ratio, where p₂ > p₁. One set of tables, taken from the Army Chemical Corps Engineering Agency for a = .05 and b = .10,  is given below: 

 

 

The ASN curve for a double sampling plan 
 

ASN Curve for a Double Sampling Plan  

Since when using a double sampling plan the sample size depends on whether or not a second sample is required, an important consideration for this kind of sampling is the Average Sample Number (ASN) curve. This curve plots the ASN versus p', the true fraction defective in an incoming lot. 

We will illustrate how to calculate the ASN curve with an example. Consider a double-sampling plan  n₁ = 50, c₁ = 2, n₂ = 100,  c₂ = 6, where n₁ is the sample size for plan 1, with accept number c₁, and n₂, c₂, are the sample size and accept number for plan 2. 

Let p' = .06. Then the chance of acceptance on the first sample which is the chance of getting two or less defectives = .416 (using binomial tables). The chance of rejection on the second sample, which is the chance of getting more than six defectives = 1-.971 = .029. The probability of making a decision on the first sample is .445, equal to the sum of .416 and .029  With complete inspection of the second sample the average sizesample is equal to the size of the first sample times the probability that there will only be one sample plus the size of the combined samples times the probability that a second sample will be necessary. For the sampling plan under consideration the average sample number (ASN) with complete inspection of the second sample for a p' of .06 is 

                                 50(.445) + 100(.555) =  106 

The general formula for an average sample number curve of a double-sampling plan with complete inspection of the second sample is 

                       ASN = n₁P₁ + (n₁ + n₂)(1 - P₁) = n₁ + n₂(1 - P₁) 

where P₁ is the probability of a decision on the first sample. The graph below shows a plot of the ASN versus p'.